fof(t21_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => k2_relat_1(k2_bintree2(A,B)) = k2_trees_2(A,B) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_bintree2,t10_bintree2,t30_finseq_2,t20_bintree2,t78_finseq_1,t18_bintree2,t78_finseq_4]),
    [file(bintree2,t21_bintree2),interesting(1.00)]).

fof(t20_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => k4_finseq_1(k2_bintree2(A,B)) = k2_finseq_1(k3_series_1(2,B)) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_finseq_1,t19_bintree2]),
    [file(bintree2,t20_bintree2),interesting(0.82)]).

fof(t19_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => k3_finseq_1(k2_bintree2(A,B)) = k3_series_1(2,B) ) ) ),
    inference(mizar_proof,[status(thm)],[t55_funct_1,d1_funct_2,d4_wellord2,d3_finseq_1,t21_card_1,t18_bintree2,t78_finseq_1]),
    [file(bintree2,t19_bintree2),interesting(0.80)]).

fof(t4_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( A = k3_finseq_2(k7_domain_1(k5_numbers,0,1))
       => k3_trees_1(A) = k1_xboole_0 ) ) ),
    inference(mizar_proof,[status(thm)],[d1_xboole_0,t3_bintree2,t2_bintree2,d1_tarski,d3_tarski,d1_tarski,d13_margrel1,t22_scmfsa_7,t23_scmfsa_7,d11_finseq_1,d12_margrel1,t31_trees_1,d8_trees_1]),
    [file(bintree2,t4_bintree2),interesting(0.79)]).

fof(t2_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree1(A) )
     => ! [B] : 
          ( m1_trees_1(B,A)
         => m2_finseq_1(B,k6_margrel1) ) ) ),
    inference(mizar_proof,[status(thm)],[t27_finseq_1,t2_xboole_1,t46_trees_1,d5_trees_2,t5_bintree1,d2_bintree1,d2_tarski,t20_finseq_2,d2_tarski,d3_tarski,d12_margrel1,d1_tarski,t55_finseq_1,t46_trees_1,t8_xboole_1,t44_finseq_1,s2_finseq_2,d4_finseq_1]),
    [file(bintree2,t2_bintree2),interesting(0.78)]).

fof(t18_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_card_1(k2_trees_2(A,B)) = k3_series_1(2,B) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_bintree2,t17_qc_lang4,t79_card_1,t29_power,t2_euclid,t5_binari_3,d12_margrel1,t6_finseq_1,d4_euclid,t70_finseq_2,t19_finseq_2,t2_euclid,t5_binari_3,d12_margrel1,d11_finseq_1,d3_tarski,t55_finseq_1,d1_tarski,d2_tarski,d4_finseq_1,t23_scmfsa_7,d11_finseq_1,t2_euclid,t59_finseq_1,d3_tarski,d6_trees_2,d3_tarski,d6_trees_2,t13_finset_1,t8_xboole_1,d3_tarski,d6_trees_2,t6_finseq_1,d3_finseq_1,t13_finseq_2,d12_margrel1,d2_tarski,d2_xboole_0,d7_xboole_0,d1_xboole_0,d3_xboole_0,t10_bintree2,d6_trees_2,d3_tarski,t55_finseq_1,d1_tarski,d2_tarski,d4_finseq_1,d12_margrel1,t23_scmfsa_7,d11_finseq_1,t19_finseq_2,t59_finseq_1,d6_trees_2,d12_margrel1,t23_scmfsa_7,d11_finseq_1,t19_finseq_2,t59_finseq_1,s3_funct_2,s3_funct_2,d1_funct_2,d1_funct_2,t20_finseq_2,d8_funct_1,t60_card_1,t45_funct_2,t22_finseq_2,t19_finseq_2,d11_finseq_1,d12_margrel1,d6_trees_2,d6_trees_2,t59_finseq_1,t16_funct_2,t81_card_1,d1_funct_2,d1_funct_2,t20_finseq_2,d8_funct_1,t60_card_1,t45_funct_2,t22_finseq_2,t19_finseq_2,d11_finseq_1,d12_margrel1,d6_trees_2,d6_trees_2,t59_finseq_1,t16_funct_2,t32_power,t30_power,t81_card_1,t53_card_2,d10_xboole_0,s1_nat_1]),
    [file(bintree2,t18_bintree2),interesting(0.78)]).

fof(t15_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => k1_funct_1(k2_bintree2(A,B),1) = k5_euclid(B) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_bintree2,t10_bintree2,d1_funct_2,t13_bintree2,t54_funct_1]),
    [file(bintree2,t15_bintree2),interesting(0.75)]).

fof(t22_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => k1_funct_1(k2_bintree2(A,1),1) = k13_binarith(k5_numbers,0) ) ),
    inference(mizar_proof,[status(thm)],[t15_bintree2,d4_euclid,t73_finseq_2]),
    [file(bintree2,t22_bintree2),interesting(0.74)]).

fof(t23_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => k1_funct_1(k2_bintree2(A,1),2) = k13_binarith(k5_numbers,1) ) ),
    inference(mizar_proof,[status(thm)],[d4_euclid,t73_finseq_2,t36_margrel1,t30_power,t16_bintree2,t15_binari_3,t36_margrel1]),
    [file(bintree2,t23_bintree2),interesting(0.73)]).

fof(t8_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( A = k3_finseq_2(k7_domain_1(k5_numbers,0,1))
      <=> ! [B] : 
            ( m1_trees_1(B,A)
           => k1_trees_2(A,B) = k7_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k5_numbers)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,0)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,1))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t3_bintree2,t4_bintree2,d2_bintree1,d3_tarski,t7_bintree2,t2_bintree2,d12_margrel1,d11_finseq_1,d10_xboole_0,d3_tarski,d11_finseq_1,t47_trees_1,d2_tarski,d2_tarski,s2_finseq_2]),
    [file(bintree2,t8_bintree2),interesting(0.68)]).

fof(t3_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( A = k3_finseq_2(k7_domain_1(k5_numbers,0,1))
       => v1_bintree1(A) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d11_finseq_1,t6_finseq_1,t19_finseq_2,d3_finseq_1,d4_finseq_4,t59_finseq_1,d2_tarski,d2_tarski,d10_xboole_0,d3_tarski,d11_finseq_1,d3_tarski,t55_finseq_1,d1_tarski,d2_tarski,d4_finseq_1,d3_tarski,t55_finseq_1,d1_tarski,d2_tarski,d4_finseq_1,t23_scmfsa_7,d11_finseq_1,t23_scmfsa_7,d11_finseq_1,d2_tarski,d5_trees_2,d2_bintree1]),
    [file(bintree2,t3_bintree2),interesting(0.67)]).

fof(t13_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => k1_funct_1(k1_bintree2(A,B),k5_euclid(B)) = 1 ) ) ),
    inference(mizar_proof,[status(thm)],[d2_bintree2,t5_binari_3,d12_margrel1,t10_bintree2,l9_bintree2,d3_finseq_5,t2_euclid,t110_finseq_2,t9_binari_3,d1_bintree2,t7_binari_3]),
    [file(bintree2,t13_bintree2),interesting(0.65)]).

fof(t16_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] : 
              ( m2_finseq_2(C,k6_margrel1,k4_finseq_2(B,k6_margrel1))
             => ( C = k5_euclid(B)
               => k1_funct_1(k2_bintree2(A,B),k3_series_1(2,B)) = k6_binarith(B,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_bintree2,t11_bintree2,d1_funct_2,t14_bintree2,t54_funct_1]),
    [file(bintree2,t16_bintree2),interesting(0.60)]).

fof(t9_bintree2,theorem,
    ( ~ v1_xboole_0(k3_finseq_2(k7_domain_1(k5_numbers,0,1)))
    & v1_trees_1(k3_finseq_2(k7_domain_1(k5_numbers,0,1))) ),
    inference(mizar_proof,[status(thm)],[t83_finseq_1,d3_tarski,d4_trees_1,d8_xboole_0,d1_trees_1,t1_bintree2,d11_finseq_1,t50_finseq_1,t50_finseq_1,t5_finseq_1,t55_finseq_1,t13_finseq_2,t57_finseq_1,d5_real_1,t38_nat_1,d2_tarski,d2_tarski,t22_scmfsa_7,t23_scmfsa_7,d11_finseq_1,d5_trees_1]),
    [file(bintree2,t9_bintree2),interesting(0.60)]).

fof(t1_bintree2,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => ! [C] : 
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ( r2_hidden(B,k3_finseq_2(A))
           => r2_hidden(k7_relat_1(B,k2_finseq_1(C)),k3_finseq_2(A)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d11_finseq_1,t23_finseq_1,d11_finseq_1]),
    [file(bintree2,t1_bintree2),interesting(0.52)]).

fof(t12_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => r1_tarski(k2_finseq_1(k3_series_1(2,B)),k2_relat_1(k1_bintree2(A,B))) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d1_finseq_1,d2_bintree2,d11_finseq_1,d12_margrel1,d3_finseq_5,t109_finseq_2,d6_trees_2,d1_funct_2,d3_finseq_5,t110_finseq_2,t11_xreal_1,t38_nat_1,t21_xreal_1,d3_binarith,d1_bintree2,t29_finseq_6,t36_binari_3,d3_binarith,d5_funct_1]),
    [file(bintree2,t12_bintree2),interesting(0.51)]).

fof(t14_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] : 
              ( m2_finseq_2(C,k6_margrel1,k4_finseq_2(B,k6_margrel1))
             => ( C = k5_euclid(B)
               => k1_funct_1(k1_bintree2(A,B),k6_binarith(B,C)) = k3_series_1(2,B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_bintree2,t11_bintree2,d3_finseq_5,t109_finseq_2,t110_finseq_2,t10_binari_3,d1_bintree2,t8_binari_3]),
    [file(bintree2,t14_bintree2),interesting(0.50)]).

fof(t10_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( A = k3_finseq_2(k7_domain_1(k5_numbers,0,1))
       => ! [B] : 
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => r2_hidden(k5_euclid(B),k2_trees_2(A,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t2_euclid,t5_binari_3,d12_margrel1,d6_trees_2]),
    [file(bintree2,t10_bintree2),interesting(0.48)]).

fof(t17_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(C,k2_finseq_1(k3_series_1(2,B)))
               => k1_funct_1(k2_bintree2(A,B),C) = k4_finseq_5(k6_margrel1,k1_binari_3(B,k5_binarith(C,1))) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t3_finseq_1,d3_finseq_5,t109_finseq_2,t110_finseq_2,d11_finseq_1,d12_margrel1,d2_bintree2,d6_trees_2,t29_finseq_6,d1_bintree2,d1_funct_2,t38_nat_1,t21_xreal_1,t50_binarith,t36_binari_3,t50_binarith,t54_funct_1]),
    [file(bintree2,t17_bintree2),interesting(0.44)]).

fof(s1_bintree2,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s1_bintree2)
       => ? [B] : 
            ( m1_subset_1(B,f1_s1_bintree2)
            & ? [C] : 
                ( m1_subset_1(C,f1_s1_bintree2)
                & p1_s1_bintree2(A,B,C) ) ) )
   => ? [A] : 
        ( v1_funct_1(A)
        & v3_trees_2(A)
        & v2_bintree1(A)
        & m3_trees_2(A,f1_s1_bintree2)
        & k1_relat_1(A) = k3_finseq_2(k7_domain_1(k5_numbers,0,1))
        & k1_funct_1(A,k1_xboole_0) = f2_s1_bintree2
        & ! [B] : 
            ( m1_trees_1(B,k1_relat_1(A))
           => p1_s1_bintree2(k3_trees_2(f1_s1_bintree2,A,B),k1_funct_1(A,k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,0))),k1_funct_1(A,k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,1)))) ) ) ),
    inference(mizar_proof,[status(thm)],[s1_wellord2,t10_mcart_1,d5_funct_1,t10_mcart_1,d1_cqc_lang,d1_cqc_lang,s2_funct_2,t7_mcart_1,t106_zfmisc_1,d1_cqc_lang,t7_mcart_1,t7_mcart_1,t7_mcart_1,t106_zfmisc_1,d1_cqc_lang,t7_mcart_1,t7_mcart_1,s9_trees_2,d3_tarski,t71_nat_1,d2_tarski,d10_xboole_0,d3_tarski,d2_tarski,d2_bintree1,d3_bintree1,d3_tarski,t71_nat_1,d2_tarski,d10_xboole_0,d3_tarski,d2_tarski,t8_bintree2]),
    [file(bintree2,s1_bintree2),interesting(0.39)]).

fof(t7_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( ! [B] : 
            ( m1_trees_1(B,A)
           => k1_trees_2(A,B) = k7_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k5_numbers)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,0)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,1))) )
       => v1_bintree1(A) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_bintree1]),
    [file(bintree2,t7_bintree2),interesting(0.38)]).

fof(t5_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( m1_bintree2(C,A)
             => ( r2_hidden(C,k2_trees_2(A,B))
               => m2_finseq_2(C,k6_margrel1,k4_finseq_2(B,k6_margrel1)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_trees_2,t110_finseq_2]),
    [file(bintree2,t5_bintree2),interesting(0.35)]).

fof(t24_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] : 
              ( ( ~ v1_xboole_0(C)
                & m2_subset_1(C,k1_numbers,k5_numbers) )
             => ( r1_xreal_0(C,k3_series_1(2,k1_nat_1(B,1)))
               => ! [D] : 
                    ( m2_finseq_2(D,k6_margrel1,k4_finseq_2(B,k6_margrel1))
                   => ( D = k1_funct_1(k2_bintree2(A,B),k3_nat_1(k1_nat_1(C,1),2))
                     => k1_funct_1(k2_bintree2(A,k1_nat_1(B,1)),C) = k7_finseq_1(D,k13_binarith(k5_numbers,k4_nat_1(k1_nat_1(C,1),2))) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t39_nat_1,t8_xreal_1,t15_nat_2,t32_power,t30_power,t27_nat_2,t3_finseq_1,t24_nat_2,t73_nat_1,t53_binarith,t23_nat_2,t73_nat_1,t53_binarith,t8_xreal_1,t15_nat_2,t16_nat_2,t53_binarith,t50_binarith,t3_finseq_1,t17_bintree2,t35_binari_3,t28_finseq_6,t17_bintree2]),
    [file(bintree2,t24_bintree2),interesting(0.19)]).

fof(s1_wellord2,theorem,
    ( ! [A] : ~ ( r2_hidden(A,f1_s1_wellord2)
        & ! [B] : ~ ( r2_hidden(B,f2_s1_wellord2)
            & p1_s1_wellord2(A,B) ) )
   => ? [A] : 
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & k1_relat_1(A) = f1_s1_wellord2
        & r1_tarski(k2_relat_1(A),f2_s1_wellord2)
        & ! [B] : 
            ( r2_hidden(B,f1_s1_wellord2)
           => p1_s1_wellord2(B,k1_funct_1(A,B)) ) ) ),
    file(wellord2,s1_wellord2),
    [interesting(0.00)]).

fof(t10_mcart_1,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(A,k2_zfmisc_1(B,C))
     => ( r2_hidden(k1_mcart_1(A),B)
        & r2_hidden(k2_mcart_1(A),C) ) ) ),
    file(mcart_1,t10_mcart_1),
    [interesting(0.00)]).

fof(d5_funct_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( B = k2_relat_1(A)
        <=> ! [C] : 
              ( r2_hidden(C,B)
            <=> ? [D] : 
                  ( r2_hidden(D,k1_relat_1(A))
                  & C = k1_funct_1(A,D) ) ) ) ) ),
    file(funct_1,d5_funct_1),
    [interesting(0.00)]).

fof(d1_cqc_lang,definition,(
    ! [A,B,C,D] : 
      ( ( A = B
       => k1_cqc_lang(A,B,C,D) = C )
      & ( A != B
       => k1_cqc_lang(A,B,C,D) = D ) ) ),
    file(cqc_lang,d1_cqc_lang),
    [interesting(0.00)]).

fof(s2_funct_2,theorem,
    ( ! [A] : 
        ( r2_hidden(A,f1_s2_funct_2)
       => r2_hidden(f3_s2_funct_2(A),f2_s2_funct_2) )
   => ? [A] : 
        ( v1_funct_1(A)
        & v1_funct_2(A,f1_s2_funct_2,f2_s2_funct_2)
        & m2_relset_1(A,f1_s2_funct_2,f2_s2_funct_2)
        & ! [B] : 
            ( r2_hidden(B,f1_s2_funct_2)
           => k1_funct_1(A,B) = f3_s2_funct_2(B) ) ) ),
    file(funct_2,s2_funct_2),
    [interesting(0.00)]).

fof(t7_mcart_1,theorem,(
    ! [A,B] : 
      ( k1_mcart_1(k4_tarski(A,B)) = A
      & k2_mcart_1(k4_tarski(A,B)) = B ) ),
    file(mcart_1,t7_mcart_1),
    [interesting(0.00)]).

fof(t106_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D))
    <=> ( r2_hidden(A,C)
        & r2_hidden(B,D) ) ) ),
    file(zfmisc_1,t106_zfmisc_1),
    [interesting(0.00)]).

fof(s9_trees_2,theorem,(
    ? [A] : 
      ( v1_funct_1(A)
      & v3_trees_2(A)
      & m3_trees_2(A,f1_s9_trees_2)
      & k1_funct_1(A,k1_xboole_0) = f2_s9_trees_2
      & ! [B] : 
          ( m1_trees_1(B,k1_relat_1(A))
         => ( k1_trees_2(k1_relat_1(A),B) = a_2_4_trees_2(A,B)
            & ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( ~ r1_xreal_0(f3_s9_trees_2(k3_trees_2(f1_s9_trees_2,A,B)),C)
                 => k1_funct_1(A,k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,C))) = k1_funct_1(f4_s9_trees_2,k4_tarski(k3_trees_2(f1_s9_trees_2,A,B),C)) ) ) ) ) ) ),
    file(trees_2,s9_trees_2),
    [interesting(0.00)]).

fof(d3_tarski,definition,(
    ! [A,B] : 
      ( r1_tarski(A,B)
    <=> ! [C] : 
          ( r2_hidden(C,A)
         => r2_hidden(C,B) ) ) ),
    file(tarski,d3_tarski),
    [interesting(0.00)]).

fof(t71_nat_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ ( ~ r1_xreal_0(2,A)
          & A != 0
          & A != 1 ) ) ),
    file(nat_1,t71_nat_1),
    [interesting(0.00)]).

fof(d2_tarski,definition,(
    ! [A,B,C] : 
      ( C = k2_tarski(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( D = A
            | D = B ) ) ) ),
    file(tarski,d2_tarski),
    [interesting(0.00)]).

fof(d10_xboole_0,definition,(
    ! [A,B] : 
      ( A = B
    <=> ( r1_tarski(A,B)
        & r1_tarski(B,A) ) ) ),
    file(xboole_0,d10_xboole_0),
    [interesting(0.00)]).

fof(d2_bintree1,definition,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( v1_bintree1(A)
      <=> ! [B] : 
            ( m1_trees_1(B,A)
           => ( ~ r2_hidden(B,k3_trees_1(A))
             => k1_trees_2(A,B) = k2_tarski(k7_finseq_1(B,k3_lang1(k1_numbers,0)),k7_finseq_1(B,k3_lang1(k1_numbers,1))) ) ) ) ) ),
    file(bintree1,d2_bintree1),
    [interesting(0.00)]).

fof(d3_bintree1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ( v2_bintree1(A)
      <=> v1_bintree1(k1_relat_1(A)) ) ) ),
    file(bintree1,d3_bintree1),
    [interesting(0.00)]).

fof(d11_finseq_1,definition,(
    ! [A,B] : 
      ( B = k13_finseq_1(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> m2_finseq_1(C,A) ) ) ),
    file(finseq_1,d11_finseq_1),
    [interesting(0.00)]).

fof(t6_finseq_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => r2_hidden(k2_xcmplx_0(A,1),k2_finseq_1(k2_xcmplx_0(A,1))) ) ),
    file(finseq_1,t6_finseq_1),
    [interesting(0.00)]).

fof(t19_finseq_2,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => k3_finseq_1(k7_finseq_1(B,k9_finseq_1(A))) = k1_nat_1(k3_finseq_1(B),1) ) ),
    file(finseq_2,t19_finseq_2),
    [interesting(0.00)]).

fof(d3_finseq_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k3_finseq_1(A)
          <=> k2_finseq_1(B) = k1_relat_1(A) ) ) ) ),
    file(finseq_1,d3_finseq_1),
    [interesting(0.00)]).

fof(d4_finseq_4,definition,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & m2_relset_1(C,A,B) )
     => ! [D] : 
          ( r2_hidden(D,k1_relat_1(C))
         => k4_finseq_4(A,B,C,D) = k1_funct_1(C,D) ) ) ),
    file(finseq_4,d4_finseq_4),
    [interesting(0.00)]).

fof(t59_finseq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => k1_funct_1(k7_finseq_1(B,k9_finseq_1(A)),k1_nat_1(k3_finseq_1(B),1)) = A ) ),
    file(finseq_1,t59_finseq_1),
    [interesting(0.00)]).

fof(t55_finseq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => ( B = k9_finseq_1(A)
      <=> ( k4_finseq_1(B) = k2_finseq_1(1)
          & k2_relat_1(B) = k1_tarski(A) ) ) ) ),
    file(finseq_1,t55_finseq_1),
    [interesting(0.00)]).

fof(d1_tarski,definition,(
    ! [A,B] : 
      ( B = k1_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> C = A ) ) ),
    file(tarski,d1_tarski),
    [interesting(0.00)]).

fof(d4_finseq_1,definition,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => ( m1_finseq_1(B,A)
      <=> r1_tarski(k2_relat_1(B),A) ) ) ),
    file(finseq_1,d4_finseq_1),
    [interesting(0.00)]).

fof(t23_scmfsa_7,theorem,(
    ! [A,B] : 
      ( m2_finseq_1(B,A)
     => ! [C] : 
          ( m2_finseq_1(C,A)
         => m2_finseq_1(k8_finseq_1(A,B,C),A) ) ) ),
    file(scmfsa_7,t23_scmfsa_7),
    [interesting(0.00)]).

fof(d5_trees_2,definition,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] : 
          ( m1_trees_1(B,A)
         => k1_trees_2(A,B) = a_2_1_trees_2(A,B) ) ) ),
    file(trees_2,d5_trees_2),
    [interesting(0.00)]).

fof(d1_xboole_0,definition,(
    ! [A] : 
      ( A = k1_xboole_0
    <=> ! [B] : ~ r2_hidden(B,A) ) ),
    file(xboole_0,d1_xboole_0),
    [interesting(0.00)]).

fof(t27_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( A = k1_xboole_0
      <=> k2_relat_1(A) = k1_xboole_0 ) ) ),
    file(finseq_1,t27_finseq_1),
    [interesting(0.00)]).

fof(t2_xboole_1,theorem,(
    ! [A] : r1_tarski(k1_xboole_0,A) ),
    file(xboole_1,t2_xboole_1),
    [interesting(0.00)]).

fof(t46_trees_1,theorem,(
    ! [A] : 
      ( m2_finseq_1(A,k5_numbers)
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & v1_trees_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( r2_hidden(k7_finseq_1(A,C),B)
               => r2_hidden(A,B) ) ) ) ) ),
    file(trees_1,t46_trees_1),
    [interesting(0.00)]).

fof(t5_bintree1,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] : 
          ( m1_trees_1(B,A)
         => ( k1_trees_2(A,B) = k1_xboole_0
          <=> r2_hidden(B,k3_trees_1(A)) ) ) ) ),
    file(bintree1,t5_bintree1),
    [interesting(0.00)]).

fof(t20_finseq_2,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C)
        & v1_finseq_1(C) )
     => ! [D] : 
          ( ( v1_relat_1(D)
            & v1_funct_1(D)
            & v1_finseq_1(D) )
         => ( k7_finseq_1(C,k9_finseq_1(A)) = k7_finseq_1(D,k9_finseq_1(B))
           => ( C = D
              & A = B ) ) ) ) ),
    file(finseq_2,t20_finseq_2),
    [interesting(0.00)]).

fof(d12_margrel1,definition,(
    k6_margrel1 = k2_tarski(0,1) ),
    file(margrel1,d12_margrel1),
    [interesting(0.00)]).

fof(t8_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,B) )
     => r1_tarski(k2_xboole_0(A,C),B) ) ),
    file(xboole_1,t8_xboole_1),
    [interesting(0.00)]).

fof(t44_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => k2_relat_1(k7_finseq_1(A,B)) = k2_xboole_0(k2_relat_1(A),k2_relat_1(B)) ) ) ),
    file(finseq_1,t44_finseq_1),
    [interesting(0.00)]).

fof(s2_finseq_2,theorem,
    ( ( p1_s2_finseq_2(k6_finseq_1(f1_s2_finseq_2))
      & ! [A] : 
          ( m2_finseq_1(A,f1_s2_finseq_2)
         => ! [B] : 
              ( m1_subset_1(B,f1_s2_finseq_2)
             => ( p1_s2_finseq_2(A)
               => p1_s2_finseq_2(k8_finseq_1(f1_s2_finseq_2,A,k12_finseq_1(f1_s2_finseq_2,B))) ) ) ) )
   => ! [A] : 
        ( m2_finseq_1(A,f1_s2_finseq_2)
       => p1_s2_finseq_2(A) ) ),
    file(finseq_2,s2_finseq_2),
    [interesting(0.00)]).

fof(d13_margrel1,definition,(
    k7_margrel1 = 0 ),
    file(margrel1,d13_margrel1),
    [interesting(0.00)]).

fof(t22_scmfsa_7,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => m2_finseq_1(k12_finseq_1(A,B),A) ) ) ),
    file(scmfsa_7,t22_scmfsa_7),
    [interesting(0.00)]).

fof(t31_trees_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C)
            & v1_finseq_1(C) )
         => ( r1_tarski(B,C)
           => r2_xboole_0(B,k7_finseq_1(C,k9_finseq_1(A))) ) ) ) ),
    file(trees_1,t31_trees_1),
    [interesting(0.00)]).

fof(d8_trees_1,definition,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( B = k3_trees_1(A)
          <=> ! [C] : 
                ( m2_finseq_1(C,k5_numbers)
               => ( r2_hidden(C,B)
                <=> ( r2_hidden(C,A)
                    & ! [D] : 
                        ( m2_finseq_1(D,k5_numbers)
                       => ~ ( r2_hidden(D,A)
                            & r2_xboole_0(C,D) ) ) ) ) ) ) ) ) ),
    file(trees_1,d8_trees_1),
    [interesting(0.00)]).

fof(t47_trees_1,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( r2_hidden(k1_xboole_0,A)
        & r2_hidden(k6_finseq_1(k5_numbers),A) ) ) ),
    file(trees_1,t47_trees_1),
    [interesting(0.00)]).

fof(t4_funct_2,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r1_tarski(k2_relat_1(B),A)
       => ( v1_funct_1(B)
          & v1_funct_2(B,k1_relat_1(B),A)
          & m2_relset_1(B,k1_relat_1(B),A) ) ) ) ),
    file(funct_2,t4_funct_2),
    [interesting(0.00)]).

fof(s2_bintree2,theorem,
    ( ( ! [A] : 
          ( m1_subset_1(A,f1_s2_bintree2)
         => ? [B] : 
              ( m1_subset_1(B,f1_s2_bintree2)
              & p1_s2_bintree2(A,B) ) )
      & ! [A] : 
          ( m1_subset_1(A,f1_s2_bintree2)
         => ? [B] : 
              ( m1_subset_1(B,f1_s2_bintree2)
              & p2_s2_bintree2(A,B) ) ) )
   => ? [A] : 
        ( v1_funct_1(A)
        & v3_trees_2(A)
        & v2_bintree1(A)
        & m3_trees_2(A,f1_s2_bintree2)
        & k1_relat_1(A) = k3_finseq_2(k7_domain_1(k5_numbers,0,1))
        & k1_funct_1(A,k1_xboole_0) = f2_s2_bintree2
        & ! [B] : 
            ( m1_trees_1(B,k1_relat_1(A))
           => ( p1_s2_bintree2(k3_trees_2(f1_s2_bintree2,A,B),k1_funct_1(A,k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,0))))
              & p2_s2_bintree2(k3_trees_2(f1_s2_bintree2,A,B),k1_funct_1(A,k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,1)))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t10_mcart_1,d1_cqc_lang,d1_cqc_lang,s1_wellord2,t4_funct_2,s9_trees_2,d3_tarski,t71_nat_1,d2_tarski,d10_xboole_0,d3_tarski,d2_tarski,d2_bintree1,d3_bintree1,d3_tarski,t71_nat_1,d2_tarski,d10_xboole_0,d3_tarski,d2_tarski,t8_bintree2,t7_mcart_1,t7_mcart_1,t7_mcart_1,t7_mcart_1]),
    [file(bintree2,s2_bintree2),interesting(0.00)]).

fof(d1_finseq_1,definition,(
    ! [A] : 
      ( v4_ordinal2(A)
     => k1_finseq_1(A) = a_1_0_finseq_1(A) ) ),
    file(finseq_1,d1_finseq_1),
    [interesting(0.00)]).

fof(d2_bintree2,definition,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( v1_bintree2(A)
      <=> A = k3_finseq_2(k7_domain_1(k5_numbers,0,1)) ) ) ),
    file(bintree2,d2_bintree2),
    [interesting(0.00)]).

fof(d3_finseq_5,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( B = k3_finseq_5(A)
          <=> ( k3_finseq_1(B) = k3_finseq_1(A)
              & ! [C] : 
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k4_finseq_1(B))
                   => k1_funct_1(B,C) = k1_funct_1(A,k2_xcmplx_0(k6_xcmplx_0(k3_finseq_1(A),C),1)) ) ) ) ) ) ) ),
    file(finseq_5,d3_finseq_5),
    [interesting(0.00)]).

fof(t109_finseq_2,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( m2_finseq_2(C,B,k4_finseq_2(A,B))
             => k3_finseq_1(C) = A ) ) ) ),
    file(finseq_2,t109_finseq_2),
    [interesting(0.00)]).

fof(d6_trees_2,definition,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_trees_2(A,B) = a_2_0_trees_2(A,B) ) ) ),
    file(trees_2,d6_trees_2),
    [interesting(0.00)]).

fof(d1_funct_2,definition,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => ( ( ( B = k1_xboole_0
           => A = k1_xboole_0 )
         => ( v1_funct_2(C,A,B)
          <=> A = k4_relset_1(A,B,C) ) )
        & ( B = k1_xboole_0
         => ( A = k1_xboole_0
            | ( v1_funct_2(C,A,B)
            <=> C = k1_xboole_0 ) ) ) ) ) ),
    file(funct_2,d1_funct_2),
    [interesting(0.00)]).

fof(t110_finseq_2,theorem,(
    ! [A,B] : 
      ( m2_finseq_1(B,A)
     => m1_subset_1(B,k4_finseq_2(k3_finseq_1(B),A)) ) ),
    file(finseq_2,t110_finseq_2),
    [interesting(0.00)]).

fof(t11_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( r1_xreal_0(A,B)
              <=> r1_xreal_0(k6_xcmplx_0(A,C),k6_xcmplx_0(B,C)) ) ) ) ) ),
    file(xreal_1,t11_xreal_1),
    [interesting(0.00)]).

fof(t38_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( ~ r1_xreal_0(k2_xcmplx_0(B,1),A)
          <=> r1_xreal_0(A,B) ) ) ) ),
    file(nat_1,t38_nat_1),
    [interesting(0.00)]).

fof(t21_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( r1_xreal_0(k2_xcmplx_0(A,B),C)
              <=> r1_xreal_0(A,k6_xcmplx_0(C,B)) ) ) ) ) ),
    file(xreal_1,t21_xreal_1),
    [interesting(0.00)]).

fof(d3_binarith,definition,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( ( r1_xreal_0(0,k6_xcmplx_0(A,B))
             => k5_binarith(A,B) = k6_xcmplx_0(A,B) )
            & ( ~ r1_xreal_0(0,k6_xcmplx_0(A,B))
             => k5_binarith(A,B) = 0 ) ) ) ) ),
    file(binarith,d3_binarith),
    [interesting(0.00)]).

fof(d1_bintree2,definition,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree1(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_trees_2(A,B),k5_numbers)
                & m2_relset_1(C,k2_trees_2(A,B),k5_numbers) )
             => ( C = k1_bintree2(A,B)
              <=> ! [D] : 
                    ( m1_bintree2(D,A)
                   => ( r2_hidden(D,k2_trees_2(A,B))
                     => ! [E] : 
                          ( m2_finseq_2(E,k6_margrel1,k4_finseq_2(B,k6_margrel1))
                         => ( E = k4_finseq_5(k6_margrel1,D)
                           => k1_funct_1(C,D) = k1_nat_1(k9_binarith(B,E),1) ) ) ) ) ) ) ) ) ),
    file(bintree2,d1_bintree2),
    [interesting(0.00)]).

fof(t29_finseq_6,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => k3_finseq_5(k3_finseq_5(A)) = A ) ),
    file(finseq_6,t29_finseq_6),
    [interesting(0.00)]).

fof(t36_binari_3,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(k3_series_1(2,A),B)
           => k9_binarith(A,k1_binari_3(A,B)) = B ) ) ) ),
    file(binari_3,t36_binari_3),
    [interesting(0.00)]).

fof(t2_euclid,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => k3_finseq_1(B) = A ) ) ),
    file(euclid,t2_euclid),
    [interesting(0.00)]).

fof(t5_binari_3,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r2_hidden(k5_euclid(A),k13_finseq_1(k6_margrel1)) ) ),
    file(binari_3,t5_binari_3),
    [interesting(0.00)]).

fof(t30_finseq_2,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m2_finseq_1(B,A)
         => ( v1_funct_1(B)
            & v1_funct_2(B,k4_finseq_1(B),A)
            & m2_relset_1(B,k4_finseq_1(B),A) ) ) ) ),
    file(finseq_2,t30_finseq_2),
    [interesting(0.00)]).

fof(t55_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => ( k2_relat_1(A) = k1_relat_1(k2_funct_1(A))
          & k1_relat_1(A) = k2_relat_1(k2_funct_1(A)) ) ) ) ),
    file(funct_1,t55_funct_1),
    [interesting(0.00)]).

fof(d4_wellord2,definition,(
    ! [A,B] : 
      ( r2_wellord2(A,B)
    <=> ? [C] : 
          ( v1_relat_1(C)
          & v1_funct_1(C)
          & v2_funct_1(C)
          & k1_relat_1(C) = A
          & k2_relat_1(C) = B ) ) ),
    file(wellord2,d4_wellord2),
    [interesting(0.00)]).

fof(t21_card_1,theorem,(
    ! [A,B] : 
      ( r2_wellord2(A,B)
    <=> k1_card_1(A) = k1_card_1(B) ) ),
    file(card_1,t21_card_1),
    [interesting(0.00)]).

fof(t17_qc_lang4,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => k2_trees_2(A,0) = k1_tarski(k1_xboole_0) ) ),
    file(qc_lang4,t17_qc_lang4),
    [interesting(0.00)]).

fof(t79_card_1,theorem,(
    ! [A] : k4_card_1(k1_tarski(A)) = 1 ),
    file(card_1,t79_card_1),
    [interesting(0.00)]).

fof(t29_power,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => k3_power(A,0) = 1 ) ),
    file(power,t29_power),
    [interesting(0.00)]).

fof(d4_euclid,definition,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k4_euclid(A) = k4_finseqop(k1_numbers,A,0) ) ),
    file(euclid,d4_euclid),
    [interesting(0.00)]).

fof(t70_finseq_2,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B,C] : 
          ( r2_hidden(B,k2_finseq_1(A))
         => k1_funct_1(k2_finseq_2(A,C),B) = C ) ) ),
    file(finseq_2,t70_finseq_2),
    [interesting(0.00)]).

fof(t13_finset_1,theorem,(
    ! [A,B] : 
      ( ( r1_tarski(A,B)
        & v1_finset_1(B) )
     => v1_finset_1(A) ) ),
    file(finset_1,t13_finset_1),
    [interesting(0.00)]).

fof(t13_finseq_2,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B,C] : 
          ( m2_finseq_1(C,B)
         => ( r2_hidden(A,k4_finseq_1(C))
           => r2_hidden(k1_funct_1(C,A),B) ) ) ) ),
    file(finseq_2,t13_finseq_2),
    [interesting(0.00)]).

fof(d2_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k2_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            | r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d2_xboole_0),
    [interesting(0.00)]).

fof(d7_xboole_0,definition,(
    ! [A,B] : 
      ( r1_xboole_0(A,B)
    <=> k3_xboole_0(A,B) = k1_xboole_0 ) ),
    file(xboole_0,d7_xboole_0),
    [interesting(0.00)]).

fof(d3_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k3_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d3_xboole_0),
    [interesting(0.00)]).

fof(s3_funct_2,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s3_funct_2)
       => ? [B] : 
            ( m1_subset_1(B,f2_s3_funct_2)
            & p1_s3_funct_2(A,B) ) )
   => ? [A] : 
        ( v1_funct_1(A)
        & v1_funct_2(A,f1_s3_funct_2,f2_s3_funct_2)
        & m2_relset_1(A,f1_s3_funct_2,f2_s3_funct_2)
        & ! [B] : 
            ( m1_subset_1(B,f1_s3_funct_2)
           => p1_s3_funct_2(B,k8_funct_2(f1_s3_funct_2,f2_s3_funct_2,A,B)) ) ) ),
    file(funct_2,s3_funct_2),
    [interesting(0.00)]).

fof(d8_funct_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
      <=> ! [B,C] : 
            ( ( r2_hidden(B,k1_relat_1(A))
              & r2_hidden(C,k1_relat_1(A))
              & k1_funct_1(A,B) = k1_funct_1(A,C) )
           => B = C ) ) ) ),
    file(funct_1,d8_funct_1),
    [interesting(0.00)]).

fof(t60_card_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( ( r1_tarski(A,k1_relat_1(B))
          & v2_funct_1(B) )
       => r2_wellord2(A,k9_relat_1(B,A)) ) ) ),
    file(card_1,t60_card_1),
    [interesting(0.00)]).

fof(t45_funct_2,theorem,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B) )
     => ( ( B = k1_xboole_0
         => A = k1_xboole_0 )
       => k2_funct_2(A,B,C,A) = k2_relat_1(C) ) ) ),
    file(funct_2,t45_funct_2),
    [interesting(0.00)]).

fof(t22_finseq_2,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m2_finseq_1(B,A)
         => ~ ( k3_finseq_1(B) != 0
              & ! [C] : 
                  ( m2_finseq_1(C,A)
                 => ! [D] : 
                      ( m1_subset_1(D,A)
                     => B != k8_finseq_1(A,C,k12_finseq_1(A,D)) ) ) ) ) ) ),
    file(finseq_2,t22_finseq_2),
    [interesting(0.00)]).

fof(t16_funct_2,theorem,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B) )
     => ( ! [D] : ~ ( r2_hidden(D,B)
            & ! [E] : ~ ( r2_hidden(E,A)
                & D = k1_funct_1(C,E) ) )
       => ( B = k1_xboole_0
          | k2_relat_1(C) = B ) ) ) ),
    file(funct_2,t16_funct_2),
    [interesting(0.00)]).

fof(t81_card_1,theorem,(
    ! [A] : 
      ( v1_finset_1(A)
     => ! [B] : 
          ( v1_finset_1(B)
         => ( r2_wellord2(A,B)
           => k4_card_1(A) = k4_card_1(B) ) ) ) ),
    file(card_1,t81_card_1),
    [interesting(0.00)]).

fof(t32_power,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( ~ r1_xreal_0(A,0)
               => k3_power(A,k2_xcmplx_0(B,C)) = k3_xcmplx_0(k3_power(A,B),k3_power(A,C)) ) ) ) ) ),
    file(power,t32_power),
    [interesting(0.00)]).

fof(t30_power,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => k3_power(A,1) = A ) ),
    file(power,t30_power),
    [interesting(0.00)]).

fof(t53_card_2,theorem,(
    ! [A] : 
      ( v1_finset_1(A)
     => ! [B] : 
          ( v1_finset_1(B)
         => ( r1_xboole_0(A,B)
           => k4_card_1(k2_xboole_0(A,B)) = k1_nat_1(k4_card_1(A),k4_card_1(B)) ) ) ) ),
    file(card_2,t53_card_2),
    [interesting(0.00)]).

fof(s1_nat_1,theorem,
    ( ( p1_s1_nat_1(0)
      & ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( p1_s1_nat_1(A)
           => p1_s1_nat_1(k1_nat_1(A,1)) ) ) )
   => ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => p1_s1_nat_1(A) ) ),
    file(nat_1,s1_nat_1),
    [interesting(0.00)]).

fof(t78_finseq_1,theorem,
    ( ! [A] : 
        ( v4_ordinal2(A)
       => k4_card_1(k2_finseq_1(A)) = A )
    & ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( k4_card_1(A) = A
          & k4_card_1(k1_card_1(A)) = A ) ) ),
    file(finseq_1,t78_finseq_1),
    [interesting(0.00)]).

fof(t78_finseq_4,theorem,(
    ! [A] : 
      ( v1_finset_1(A)
     => ! [B] : 
          ( v1_finset_1(B)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,A,B)
                & m2_relset_1(C,A,B) )
             => ( ( k4_card_1(A) = k4_card_1(B)
                  & v2_funct_1(C) )
               => k2_relat_1(C) = B ) ) ) ) ),
    file(finseq_4,t78_finseq_4),
    [interesting(0.00)]).

fof(l9_bintree2,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m2_finseq_1(B,A)
         => m2_finseq_1(k4_finseq_5(A,B),A) ) ) ),
    file(bintree2,l9_bintree2),
    [interesting(0.00)]).

fof(t9_binari_3,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k4_finseq_5(k1_numbers,k5_euclid(A)) = k5_euclid(A) ) ),
    file(binari_3,t9_binari_3),
    [interesting(0.00)]).

fof(t7_binari_3,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] : 
          ( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
         => ( B = k5_euclid(A)
           => k9_binarith(A,B) = 0 ) ) ) ),
    file(binari_3,t7_binari_3),
    [interesting(0.00)]).

fof(t54_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => ! [B] : 
            ( ( v1_relat_1(B)
              & v1_funct_1(B) )
           => ( B = k2_funct_1(A)
            <=> ( k1_relat_1(B) = k2_relat_1(A)
                & ! [C,D] : 
                    ( ( ( r2_hidden(C,k2_relat_1(A))
                        & D = k1_funct_1(B,C) )
                     => ( r2_hidden(D,k1_relat_1(A))
                        & C = k1_funct_1(A,D) ) )
                    & ( ( r2_hidden(D,k1_relat_1(A))
                        & C = k1_funct_1(A,D) )
                     => ( r2_hidden(C,k2_relat_1(A))
                        & D = k1_funct_1(B,C) ) ) ) ) ) ) ) ) ),
    file(funct_1,t54_funct_1),
    [interesting(0.00)]).

fof(t73_finseq_2,theorem,(
    ! [A] : k2_finseq_2(1,A) = k9_finseq_1(A) ),
    file(finseq_2,t73_finseq_2),
    [interesting(0.00)]).

fof(t36_margrel1,theorem,
    ( k7_margrel1 = 0
    & k8_margrel1 = 1 ),
    file(margrel1,t36_margrel1),
    [interesting(0.00)]).

fof(t11_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( A = k3_finseq_2(k7_domain_1(k5_numbers,0,1))
       => ! [B] : 
            ( ( ~ v1_xboole_0(B)
              & m2_subset_1(B,k1_numbers,k5_numbers) )
           => ! [C] : 
                ( m2_finseq_2(C,k6_margrel1,k4_finseq_2(B,k6_margrel1))
               => r2_hidden(C,k2_trees_2(A,B)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t109_finseq_2,d11_finseq_1,d12_margrel1,d6_trees_2]),
    [file(bintree2,t11_bintree2),interesting(0.00)]).

fof(t10_binari_3,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
         => ( B = k5_euclid(A)
           => k4_finseq_5(k6_margrel1,k6_binarith(A,B)) = k6_binarith(A,B) ) ) ) ),
    file(binari_3,t10_binari_3),
    [interesting(0.00)]).

fof(t8_binari_3,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] : 
          ( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
         => ( B = k5_euclid(A)
           => k9_binarith(A,k6_binarith(A,B)) = k5_real_1(k3_series_1(2,A),1) ) ) ) ),
    file(binari_3,t8_binari_3),
    [interesting(0.00)]).

fof(t15_binari_3,theorem,(
    k6_binarith(1,k13_binarith(k6_margrel1,k7_margrel1)) = k13_binarith(k6_margrel1,k8_margrel1) ),
    file(binari_3,t15_binari_3),
    [interesting(0.00)]).

fof(t39_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(1,A)
       => A = 0 ) ) ),
    file(nat_1,t39_nat_1),
    [interesting(0.00)]).

fof(t8_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( r1_xreal_0(A,B)
              <=> r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,C)) ) ) ) ) ),
    file(xreal_1,t8_xreal_1),
    [interesting(0.00)]).

fof(t15_nat_2,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( r1_xreal_0(A,B)
           => ( r1_xreal_0(A,0)
              | r1_xreal_0(1,k3_nat_1(B,A)) ) ) ) ) ),
    file(nat_2,t15_nat_2),
    [interesting(0.00)]).

fof(t27_nat_2,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(B,k2_nat_1(2,A))
           => r1_xreal_0(k3_nat_1(k1_nat_1(B,1),2),A) ) ) ) ),
    file(nat_2,t27_nat_2),
    [interesting(0.00)]).

fof(t3_finseq_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( r2_hidden(A,k2_finseq_1(B))
          <=> ( r1_xreal_0(1,A)
              & r1_xreal_0(A,B) ) ) ) ) ),
    file(finseq_1,t3_finseq_1),
    [interesting(0.00)]).

fof(t24_nat_2,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ v1_abian(A)
      <=> k4_nat_1(A,2) = 1 ) ) ),
    file(nat_2,t24_nat_2),
    [interesting(0.00)]).

fof(t73_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ! [C] : 
              ( v4_ordinal2(C)
             => k4_nat_1(A,B) = k4_nat_1(k2_xcmplx_0(k3_xcmplx_0(B,C),A),B) ) ) ) ),
    file(nat_1,t73_nat_1),
    [interesting(0.00)]).

fof(t53_binarith,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( r1_xreal_0(B,A)
           => k2_xcmplx_0(k5_binarith(A,B),B) = A ) ) ) ),
    file(binarith,t53_binarith),
    [interesting(0.00)]).

fof(t23_nat_2,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( v1_abian(A)
      <=> k4_nat_1(A,2) = 0 ) ) ),
    file(nat_2,t23_nat_2),
    [interesting(0.00)]).

fof(t16_nat_2,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( A != 0
           => k3_nat_1(k2_xcmplx_0(B,A),A) = k1_nat_1(k3_nat_1(B,A),1) ) ) ) ),
    file(nat_2,t16_nat_2),
    [interesting(0.00)]).

fof(t50_binarith,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( r1_xreal_0(A,B)
           => k5_binarith(B,A) = k6_xcmplx_0(B,A) ) ) ) ),
    file(binarith,t50_binarith),
    [interesting(0.00)]).

fof(t35_binari_3,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_binari_3(k1_nat_1(A,1),B) = k7_finseq_1(k13_binarith(k1_numbers,k4_nat_1(B,2)),k1_binari_3(A,k3_nat_1(B,2))) ) ) ),
    file(binari_3,t35_binari_3),
    [interesting(0.00)]).

fof(t28_finseq_6,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => k3_finseq_5(k7_finseq_1(k9_finseq_1(A),B)) = k7_finseq_1(k3_finseq_5(B),k9_finseq_1(A)) ) ),
    file(finseq_6,t28_finseq_6),
    [interesting(0.00)]).

fof(t6_bintree2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( ! [B] : 
            ( m1_trees_1(B,A)
           => k1_trees_2(A,B) = k7_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k5_numbers)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,0)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,1))) )
       => k3_trees_1(A) = k1_xboole_0 ) ) ),
    inference(mizar_proof,[status(thm)],[d1_xboole_0,t5_bintree1]),
    [file(bintree2,t6_bintree2),interesting(0.00)]).

fof(t83_finseq_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => r1_tarski(k13_finseq_1(A),k13_finseq_1(B)) ) ),
    file(finseq_1,t83_finseq_1),
    [interesting(0.00)]).

fof(d4_trees_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( B = k1_trees_1(A)
        <=> ! [C] : 
              ( r2_hidden(C,B)
            <=> ? [D] : 
                  ( v1_relat_1(D)
                  & v1_funct_1(D)
                  & v1_finseq_1(D)
                  & C = D
                  & r2_xboole_0(D,A) ) ) ) ) ),
    file(trees_1,d4_trees_1),
    [interesting(0.00)]).

fof(d8_xboole_0,definition,(
    ! [A,B] : 
      ( r2_xboole_0(A,B)
    <=> ( r1_tarski(A,B)
        & A != B ) ) ),
    file(xboole_0,d8_xboole_0),
    [interesting(0.00)]).

fof(d1_trees_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( r1_tarski(A,B)
          <=> ? [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & A = k7_relat_1(B,k2_finseq_1(C)) ) ) ) ) ),
    file(trees_1,d1_trees_1),
    [interesting(0.00)]).

fof(t23_finseq_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B,C] : 
          ( m2_finseq_1(C,B)
         => m2_finseq_1(k7_relat_1(C,k2_finseq_1(A)),B) ) ) ),
    file(finseq_1,t23_finseq_1),
    [interesting(0.00)]).

fof(t50_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] : 
              ( m2_finseq_1(k7_finseq_1(A,B),C)
             => ( m2_finseq_1(A,C)
                & m2_finseq_1(B,C) ) ) ) ) ),
    file(finseq_1,t50_finseq_1),
    [interesting(0.00)]).

fof(t5_finseq_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ( A = 0
        | r2_hidden(A,k2_finseq_1(A)) ) ) ),
    file(finseq_1,t5_finseq_1),
    [interesting(0.00)]).

fof(t57_finseq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => ( B = k9_finseq_1(A)
      <=> ( k3_finseq_1(B) = 1
          & k1_funct_1(B,1) = A ) ) ) ),
    file(finseq_1,t57_finseq_1),
    [interesting(0.00)]).

fof(d5_real_1,definition,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
          <=> ~ ( r1_xreal_0(B,A)
                & B != A ) ) ) ) ),
    file(real_1,d5_real_1),
    [interesting(0.00)]).

fof(d5_trees_1,definition,(
    ! [A] : 
      ( v1_trees_1(A)
    <=> ( r1_tarski(A,k13_finseq_1(k5_numbers))
        & ! [B] : 
            ( m2_finseq_1(B,k5_numbers)
           => ( r2_hidden(B,A)
             => r1_tarski(k1_trees_1(B),A) ) )
        & ! [B] : 
            ( m2_finseq_1(B,k5_numbers)
           => ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( ( r2_hidden(k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,C)),A)
                        & r1_xreal_0(D,C) )
                     => r2_hidden(k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,D)),A) ) ) ) ) ) ) ),
    file(trees_1,d5_trees_1),
    [interesting(0.00)]).